Properties

Label 1-5e2-25.4-r0-0-0
Degree $1$
Conductor $25$
Sign $0.968 + 0.248i$
Analytic cond. $0.116099$
Root an. cond. $0.116099$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s − 7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)12-s + (0.809 − 0.587i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s − 18-s + (0.309 − 0.951i)19-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s − 7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)12-s + (0.809 − 0.587i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s − 18-s + (0.309 − 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.968 + 0.248i$
Analytic conductor: \(0.116099\)
Root analytic conductor: \(0.116099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 25,\ (0:\ ),\ 0.968 + 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8746047634 + 0.1104882761i\)
\(L(\frac12)\) \(\approx\) \(0.8746047634 + 0.1104882761i\)
\(L(1)\) \(\approx\) \(1.137110429 + 0.1227687919i\)
\(L(1)\) \(\approx\) \(1.137110429 + 0.1227687919i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−38.566281692971408793603972926856, −37.701827944379050308506558826541, −35.95065728837175317528230347978, −34.02178254415774019504554443668, −33.08680663122373125628669968924, −31.993422626565167957055373117046, −30.94902972217478728491446142613, −29.03141770700617946926343968269, −28.537804100970896404196821387387, −26.88689509289986965592410580026, −25.350915097664899028332910338267, −23.28624379966554812817764368891, −22.62593152657361199528143198695, −21.21288945616132045470134485323, −20.26784925989538997233437077417, −18.58274875785159250852461552240, −16.32069126609366156415811923032, −15.34497249886866747459541622087, −13.67535680257078377851095147170, −12.11091820547122992104187370874, −10.61201408560344711140493055722, −9.46508246406389451808855605233, −6.33072922106933373242898651201, −4.720844063769540651069167836267, −3.12335403531942608139244869971, 3.06516421169049963067569008286, 5.58061874668819666636236559749, 6.781405800920544365901057691297, 8.33490915515581935912059211202, 11.15123966521173621738261783094, 12.891511748596364438957922994, 13.44239159278436398126373315317, 15.45222729004562975582254322076, 16.77292078493330666291596245636, 18.24473803468583995215885283700, 19.86038415952044778886940457548, 21.736099922508683650078333968298, 23.03346078159664038100689143292, 23.894149400292464899067347399555, 25.255789904816721094420869637168, 26.23136205482279920852304648021, 28.58024322793023298395798562663, 29.70781264532335625179810739401, 30.840750783250949686215772805529, 32.06619796099978803452237714237, 33.35389369815977330575069087417, 34.91325396356181924805760772548, 35.31476693130028434345842290738, 36.96413157370548067780585986765, 38.979590011150217333522778791294

Graph of the $Z$-function along the critical line